1. Field of the Invention
The present invention relates to a counter-torque device of a helicopter.
2. Background
Nonuniform spacing of fan blades to provide reduced noise levels and the redistribution of the frequencies at which there is noise energy so as to generate fewer perceptible sounds is disclosed in “Noise Reduction by Applying Modulation Principles,” by Donald Ewald et al., published in the Journal of the Acoustical Society of America, Vol. 49, No. 5 (Part I), 1971, pp. 1381-1385, the entire disclosure of which is incorporated herein by reference thereto. The Ewald et al. article states the following with respect to the modulated positions between fan blades:
Modulated positions between fan blades are determined by the sinusoidal equation θi′=θi+Δθ sin(mθi), where θi is the ith blade position
Modulated positions are described by:θi′=θi+Δθ sin(mθi),  (1)where θi is the ith blade position in an evenly spaced fan arrangement, θi′ is the ith blade position after rearranging the blades, Δθ is some maximum blade-angle change (the modulation amplitude), and m is the number of times the modulation cycle is repeated in one revolution of the fan . . . .The noise resulting from sinusoidal modulation of the fundamental blade passing tone may be expressed by the classical sinusoidal phase-modulation equationF(t)=A0 sin(2πF0t+Δφ sin 2πvt),  (3)where A0 is amplitude of the fundamental blade passing tone; F0=If2, blade passing frequency; I, number of blades; f2, shaft rotational frequency; v=mf8, the modulation frequency; and Δφ=IΔθ, phase-modulation amplitude.
Δφ refers to an angle which goes from zero to 2π throughout each nominal blade spacing, and Δθ is an angle which goes from zero to 2π for each revolution of the shaft. This means that Δφ will go from zero to 2πI times for every time that Δθ goes from zero to 2π . . . .
By using the trigonometric relationssin p cos q=1/2[ sin(p+q)+sin(p−q)],sin(p+q=sin p+cos q)+cos p sin q and the relations between the Bessel and trigonometric functions
            cos      ⁡              (                  p          ⁢                                          ⁢          sin          ⁢                                          ⁢          q                )              =                            J          0                ⁡                  (          p          )                    +              2        ⁢                              ∑                          n              =              1                        ∞                    ⁢                      [                                                            J                                      2                    ⁢                    n                                                  ⁡                                  (                  p                  )                                            ⁢                              cos                ⁡                                  (                                      2                    ⁢                    nq                                    )                                                      ]                                ,          ⁢            sin      ⁡              (                  p          ⁢                                          ⁢          sin          ⁢                                          ⁢          q                )              =          2      ⁢                        ∑                      n            =            1                    ∞                ⁢                  [                                                    J                                                      2                    ⁢                    n                                    -                  1                                            ⁡                              (                p                )                                      ⁢                          sin              ⁡                              (                                                      2                    ⁢                    n                                    -                  1                                )                                      ⁢            q                    ]                      ,where Jn(p) is the Bessel function of the first kind, order n, argument p, it can be shown that
                              f          ⁡                      (            t            )                          =                              A            0                    ⁢                                    {                                                                                          J                      0                                        ⁡                                          (                                              >                        ϕ                                            )                                                        ⁢                                      sin                    ⁡                                          (                                              2                        ⁢                                                                                                  ⁢                        π                        ⁢                                                                                                  ⁢                                                  F                          0                                                ⁢                        t                                            )                                                                      +                                                      ∑                                          n                      =                      1                                        ∞                                    ⁢                                                                                    J                        n                                            ⁡                                              (                                                  Δ                          ⁢                                                                                                          ⁢                          ϕ                                                )                                                              ⁢                                          sin                      ⁡                                              [                                                  2                          ⁢                                                                                                          ⁢                                                      π                            ⁡                                                          (                                                                                                F                                  0                                                                +                                nv                                                            )                                                                                ⁢                          t                                                ]                                                                                            +                                                      ∑                                          n                      =                      1                                        ∞                                    ⁢                                                                                    (                                                  -                          1                                                )                                            n                                        ⁢                                                                  J                        n                                            ⁡                                              (                        Δϕ                        )                                                              ⁢                                          sin                      ⁡                                              [                                                  2                          ⁢                                                      π                            ⁡                                                          (                                                                                                F                                  0                                                                +                                nv                                                            )                                                                                ⁢                          t                                                ]                                                                                                        }                        .                                              (        4        )            
Equation 4 illustrates that the frequency spectrum will consist of a center frequency at F0 with an amplitude of A0J0(Δφ) and a number of side bands at integer multiples of v from the center frequency, with amplitudes symmetric about the center frequency.
The values of Jn(Δφ) may be found in many mathematical handbooks and are shown graphically in FIG. 2(a) [which is FIG. 8(a) in the subject application].
An example of how the frequency spectrum may be determined for a given Δφ is shown in FIG. 2 [FIG. 8(b) in the subject application]. A trial value is chosen for Δφ. Then a vertical line is drawn through the trial value of Δφ. The intersection of this line with the Jn(Δφ) curves indicates the relative amplitudes of the resulting components at frequencies F0±nv. The resulting frequency spectrum, FIG. 2(b) [which is FIG. 8(b) in the subject application] is given to the right of the graph. Note that absolute values are plotted on the frequency spectrum shown in FIG. 2(b). The dashed lines in FIG. 2(b) indicate the normalized amplitude of the fundamental blade passing frequency tone for a fan with evenly spaced blades . . . .
The Bessel series is for a continuous phase-modulated function, while the actual frequency spectrum of the fan is produced by a number of more nearly discrete events. The amplitudes in the frequency spectrum obtained from the Bessel series will therefore differ somewhat from those obtained from the fan. The Bessel series, however, will more closely approximate the actual fan spectrum when the number of blades is large.
In order to increase acoustic performance, it is known to have the blades of a rotor that rotates in a transverse duct have an angular distribution according to an uneven azimuth modulation given by the known sinusoidal law θn=n×360°/b+Δθ sin(m×n×360°/b) where θn is the angular position of the nth of the blades counted in series from an arbitrary origin, b is the number of blades, m is modulation factor being a whole number chosen from 1 to 4, which is not prime with the number b of blades, chosen from 6 to 12, and Δθ is a constant chosen to be greater than or equal to a minimum value Δθmin, which is such that the product Δθmin times b is chosen within a range of values extending from 1.5 radian to 1 radian, such as set forth in U.S. Pat. No. 5,566,907 to Marze et al., the entire disclosure of which is incorporated herein by reference thereto. However, such a method does not result in a balanced rotor wherein modulation factor m is selected to be prime with the number of blades, including where modulation factor m=1.
Additionally, in both the Ewald et al. article and U.S. Pat. No. 5,566,907, Δθ is a constant. The methods of the prior art as disclosed in the above-mentioned Ewald et al. article and U.S. Pat. No. 5,566,907, with a constant Δθ, do not result in a balanced rotor when modulation factor m=1 and do not result in a balanced rotor when modulation factor m=2 for an odd number of blades. Additionally, as seen in the Ewald et al. article, for any given Δφ, at most two Bessel functions (Jn), as seen in FIG. 2 of the Ewald et al. article (FIG. 8(a) of the subject application), will have the same value. Thus, the fundamental harmonic (amplitude determined by J0) and the harmonics on either side (determined by J1) will have approximately the same amplitude for Δφ, that is Δφ=1.5, which is also consistent with U.S. Pat. No. 5,566,907.